Magnetohydrodynamics (MHD)

Intuition

When a plasma is dense and cool enough that particle collisions are frequent, we can forget individual particle orbits and treat the plasma as a single conducting fluid. This is magnetohydrodynamics (MHD). It is the workhorse theory for fusion reactors, solar flares, and interstellar dynamics.

Formal Definition

MHD is the one-fluid theory obtained by taking the first few velocity moments of the Vlasov equation and imposing a scalar pressure and Ohm’s law closure.

Mathematical Formulation

Continuity:

\[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf{U}) = 0\]

Momentum (Navier–Stokes):

\[\rho\frac{D\mathbf{U}}{Dt} = \mathbf{J}\times\mathbf{B} - \nabla p + \mu \nabla^2 \mathbf{U}\]

Induction:

\[\frac{\partial \mathbf{B}}{\partial t} = \nabla\times(\mathbf{U}\times\mathbf{B}) + \eta \nabla^2 \mathbf{B}\]

Ohm’s law:

\[\mathbf{E} + \mathbf{U}\times\mathbf{B} = \eta \mathbf{J}\]

Derivation

Continuity: Integrate the zeroth velocity moment of the Boltzmann equation. The collisionless form becomes the fluid continuity equation.
Momentum: Multiply Boltzmann by \(m\mathbf{v}\) and integrate. The pressure tensor is closed by \(P_{ij} \to p \delta_{ij}\) (isotropic pressure).
Induction: Use \(\mathbf{E} + \mathbf{U}\times\mathbf{B} = \eta\mathbf{J}\) and Faraday’s law \(\partial \mathbf{B}/\partial t = -\nabla\times\mathbf{E}\).
Closure: Assume the equation of state \(p = n k_B T\) and an isentropic or isothermal relation.

Worked Example

Z-pinch equilibrium: A cylindrical plasma column carries current \(I\) along \(\hat{z}\). The azimuthal magnetic field is \(B_\theta = \mu_0 I/(2\pi r)\). The Lorentz force is inward: \(\mathbf{J}\times\mathbf{B} = -(B_\theta^2/\mu_0)\hat{r}\). For force balance with pressure gradient \(dp/dr = -B_\theta^2/\mu_0\): Integrate to find \(p(r) = p_0 - (\mu_0 I^2/8\pi^2)\ln(r)\). This profile shows that peak pressure is at the axis and falls off logarithmically.

Common Mistakes

Treating MHD as always valid. MHD breaks down when \(\rho \to 0\), when kinetic effects matter, or when the Hall term is important.
Neglecting energy equation. Stating \(p\) without its evolution equation loses information about heating and cooling.
Assuming ideal MHD in reconnection. Ideal MHD forbids reconnection; resistivity \(\eta\) is essential.

Quiz Questions

Conceptual: Why does the induction equation look like the advection of a passive scalar?
Computational: For a tokamak with \(n = 10^{20}\,\text{m}^{-3}\), \(T = 1\) keV, and \(B = 2\) T, estimate \(v_A\).
MCQ: In ideal MHD (\(\eta=0\)), what happens to magnetic field lines?
A) Diffuse through the plasma
B) Move with the fluid (frozen-in)
C) Decay exponentially
D) Become parallel to velocity